Concept

Example 1 Use the Angle Addition Postulate CONSTRUCTION Using a protractor, a construction worker measures that the angle a beam makes with a ceiling is 42°. What is the measure of the angle the beam makes with the wall? The ceiling and the wall make a 90  angle. Let  1 be the angle between the beam and the ceiling. Let  2 be the angle between the beam and the wall. m  1 + m  2= 90Angle Addition Postulate 42 + m  2= 90m  1 = – 42 + m  2= 90 – 42Subtraction Property of Equality m  2= 48Substitution

Example 1 Use the Angle Addition Postulate Answer:The beam makes a 48° angle with the wall.

A.A B.B C.C D.D Example 1 A.32 B.94 C.104 D.116 Find m  1 if m  2 = 58 and m  JKL = 162. J 2 K L 1 M

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Example 2 Use Supplement or Complement TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. When the second hand bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? UnderstandMake a sketch of the situation. The time is 4 o’clock and the second hand bisects the angle between the hour and minute hands.

= 120 Example 2 Use Supplement or Complement PlanUse the Angle Addition Postulate and the definition of angle bisector. SolveSince the angles are congruent by the definition of angle bisector, each angle is 60°. Answer:Both angles are 60°. CheckUse the Angle Addition Postulate to check your answer. m  1 + m  2 = 120

A.A B.B C.C D.D Example 2 A.20 B.30 C.40 D.50 QUILTING The diagram below shows one square for a particular quilt pattern. If m  BAC = m  DAE = 20, and  BAE is a right angle, find m  CAD.

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Example 3 Proofs Using Congruent Comp. or Suppl. Theorems Given: Prove:

Example 3 Proofs Using Congruent Comp. or Suppl. Theorems 1. Given 1.m  3 + m  1 = 180  1 and  4 form a linear pair. 4.  s suppl. to same  are . 4.  3   4 Proof: StatementsReasons 2. Linear pairs are supplementary. 2.  1 and  4 are supplementary. 3. Definition of supplementary angles 3.  3 and  1 are supplementary.

Example 3 In the figure,  NYR and  RYA form a linear pair,  AXY and  AXZ form a linear pair, and  RYA and  AXZ are congruent. Prove that  NYR and  AXY are congruent.

Example 3 Which choice correctly completes the proof? Proof: StatementsReasons 1. Given 1. linear pairs. 2.If two  s form a linear pair, then they are suppl.  s Given  NYR   AXY 4. ____________ ?

A.A B.B C.C D.D Example 3 A.Substitution B.Definition of linear pair C.  s supp. to the same  or to   s are . D.Definition of supplementary  s

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Example 4 Use Vertical Angles If  1 and  2 are vertical angles and m  1 = d – 32 and m  2 = 175 – 2d, find m  1 and m  2.  1  2Vertical Angles Theorem m  1=m  2Definition of congruent angles d – 32=175 – 2dSubstitution 3d – 32=175Add 2d to each side. 3d=207Add 32 to each side. d=69Divide each side by 3.

Example 4 Use Vertical Angles Answer: m  1 = 37 and m  2 = 37 m  1=d – 32m  2 = 175 – 2d =69 – 32 or 37= 175 – 2(69) or 37

A.A B.B C.C D.D Example 4 A. B. C. D.

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